Periodic orbits in complex Abel equations |
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Authors: | Anna Cima Francesc Mañosas |
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Institution: | Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain |
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Abstract: | This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z3+B(t)z2+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z3+B(t)z2 studied in the literature, where the center variety is located in a finite number of connected components. |
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Keywords: | primary 34C07 secondary 34C23 34C25 37C27 |
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